Double-Critical Hadwiger Conjecture

A connected graph is double-critical $k$-chromatic if it has chromatic number $k$ and, for every edge $uv$, deleting $u$ and $v$ lowers the chromatic number by $2$. Double-Critical Hadwiger Conjecture. Every double-critical $k$-chromatic graph contains a $K_k$ minor. The conjecture is proved by the cited authors for $k\leq 7$, while the case $k=8$ is open in the source.